Let \(\mathcal X\) and \({\mathcal Y}\) be infinite-dimensional complex Banach spaces. The author classifies continuous linear surjections \(\varphi\) on \(B({\mathcal X})\), which are simultaneously locally spectrally bounded from above and from below at a given nonzero vector. More precisely, let \(r_T(x):=\liminf_{n\to\infty}\|T^nx\|^{1/n}\) be a local spectral radius of an operator \(T\). If, for some positive constants \(m\leq M\) and a pair of nonzero vectors \((x_0,y_0)\in{\mathcal X}\times {\mathcal X}\),
\[ m\, r_T(x_0)\leq r_{\varphi(T)}(y_0)\leq M \,r_T(x_0),\quad T\in B({\mathcal X}),\tag{1} \]
then \(\varphi\) is a scalar multiple of an inner isomorphism given by a similarity which maps \(x_0\) to \(y_0\). A similar result is given for continuous linear surjections \(\varphi\) from \(B({\mathcal X})\) onto \(B({\mathcal Y})\) when the local spectral radius \(r_-(.)\) is replaced by an inner local spectral radius.
It is further shown that, under additional assumptions on a space or on a map, it suffices to consider only one of the two inequalities in (1) and obtain the same classification. This holds, for example, in the case when \({\mathcal X}\) and \({\mathcal Y}\) are Hilbert spaces, or in the case when \(r_-(.)\) is replaced by an inner local spectral radius and a map is unital. Similar results are obtained for continuous linear maps which compress or expand the local spectrum at fixed nonzero vectors.